A remark on a 3-fold constructed by Colliot-Thélène and Voisin (Q2183382)

Let \(X\) be a smooth complex projective variety and let \(\mathrm{CH}^{p}(X)_{\mathrm{hom}}\rightarrow J^{p}(X)\) be the \(p^{\mathrm{th}}\) Abel-Jacobi map. We denote by \(A^{p}(X)\) the quotient of \(\mathrm{CH}^{p}(X)\) by the group of algebraically trivial cycles, so that we have an induced map \[ \psi^{p}:A^{p}(X)\rightarrow J_{a}^{p}(X), \] where \(J_{a}^{p}(X)\) is the image of the \(p^{\mathrm{th}}\) Abel-Jacobi map. The author deals with the following problem. Suppose \(\phi:A^{p}(X)\rightarrow A\) is a regular homomorphism, where \(A\) is an abelian variety. Is it true that \(\phi\) factors through \(\psi^{p}\), i.e., that \(\psi^{p}\) is universal? The answer is open for \(3\le p\le\dim X\), in other cases is known to be affirmative. The main result of the paper under review is a sufficient condition for \(\psi^{p}\) to not be universal. Moreover, suppose that \(Y\) is a smooth projective variety such that \(CH_{0}(Y)\) is supported on a \(3\)-dimensional closed subset. The author proves that if \(p\in\{3,\dim Y-1\}\), then \(\psi^{p}\) is universal if and only if \(\psi_{\mathrm{tors}}^{p}:A^{p}(X)_{tors}\rightarrow J_{a}^{p}(X)_{\mathrm{tors}}\) is an isomorphism. The most important application is the following. Let \(V\) be the \(3\)-fold constructed by \textit{J.-L. Colliot-Thélène} and \textit{C. Voisin} [Duke Math. J. 161, No. 5, 735--801 (2012; Zbl 1244.14010)]. The author proves that if the generalized Bloch conjecture holds for \(V\), then there exists an elliptic curve \(E\) such that \(\psi^{3}:A^{3}(V\times E)\rightarrow J_{a}^{3}(V\times E)\) is not universal. In the last section, the author analyzes other possible applications of his theorem.